According to Centers for Disease Control and Prevention, in 2015-2016, more than 12% of adults at the age of 20 and above in the United States have cholesterol levels higher than 240mg/ml. A high cholesterol level is one of the major risk factors for coronary heart disease, heart attack and stroke. While there are two types of cholesterol, low-density lipoprotein (LDL) is known as the “bad” one that leads to the buildup of cholesterol in arteries. The goal of this project is to explore factors that can affect the level of LDL in a human body, and the result is intended to help readers gain insights on how to balance the level of LDL in order to control/prevent cardiovascular diseases.
The analysis result shows that …
The data analysis is performed with the NHANES 2015-2016 data. The dependent variable, the level of low-density lipoprotein (LDL), measured by mg/dl, is selected from the dataset Cholesterol - LDL & Triglycerides of the laboratory data. Since a high level of triglyceride is believed to be associated with a high level of LDL, it is included in the model as the independent variable. Triglyceride data is also obtained from Cholesterol - LDL & Triglycerides of the laboratory data. We also include blood pressure readings from Blood Pressure dataset of the examination data. According to the data description, some participants have multiple blood pressure readings. For simplicity, we use averaged systolic and diastolic blood pressure readings as blood pressure measurements for each individual. Averaged intakes of fat and cholesterol, computed using both the First and the Second Day Total Nutrient Intakes of the dietary data, are added as independent variables as well. To account for more individual differences, we also include gender, race and age from the Demographics Data, and height, weight and BMI information from Body Measures of the examination data as additional covariates. Furthermore, SEQN, the respondent sequence number, is utilized as the unique identifier to match responses for each respondent. Finally, we removed all rows containing missing values, and there are a total of 2503 observations available for further analysis.
We fit models using multiple linear regression techniques and then perform model selections to choose the model that best describes the level of LDL. For the very first model, we regress the dependent variable LDL on all predictors:
\(\mathbf{LDL}\) ~ \(\mathbf{age + race + gender + height + weight + BMI + fat + cholesterol + triglyceride + diastolic + systolic}\) (1)
Note that covariates gender and race are treated as categorical variables.
A check of the relationship between residuals and fitted values suggests a transformation for the dependent variable (shows non-linearity).
Residual plot of the full model
With the help of the Box-Cox test:
Box-Cox Transformation plot
we identify that the square root transformation is the best choice.
We then fit a new linear model with the transformed dependent variable, \(\sqrt{LDL}\):
\(\mathbf{\sqrt{LDL}}\) ~ \(\mathbf{age + race + gender + height + weight + BMI + fat + cholesterol + triglycerides + diastolic + systolic}\) (2)
As the model has as many as 11 covariates and some of them are insignificant under t-test, we then use the stepwise selection technique to choose variables that best explain \(\sqrt{LDL}\).
Besides, we consider transformations upon predictors. Considering partial residual plots:
Partial Residual Plot
we find out that variables age and triglycerides violate the linear structure assumption. Both of these plots exhibit a quadratic form, so in addition to response variables in the full model (2) above, we add age2 and triglycerides2 to the linear regression model:
\(\mathbf{\sqrt{LDL}}\) ~ \(\mathbf{age + age^2 + race + gender + height + weight + BMI + fat + cholesterol + triglycerides + triglycerides^2 + diastolic + systolic}\) (3)
Just as the process before, we will execute the stepwise model selection technique to choose the most significant variables of this model.
Steps outlined above are carried out in R, Stata and Python. In R, we use the package data.table for data cleaning.
Note that for R code, we will use data.table() package to mutating the data, and after mutating, we will focus on the analysis.
## Group Project HTML
## Author: Huayu Li, huayuli@umich.edu
## Updated: Dec. 8 2019
#### Data cleaning using data.table
## Libraries: -------------------------------------------------------------------------
library(data.table)
library(foreign)
library(tidyverse)
## 80: --------------------------------------------------------------------------------
## Read the datasets
demo=data.table(read.xport("./Original Data/DEMO_I.XPT.txt"))
tot1=data.table(read.xport("./Original Data/DR1TOT_I.XPT.txt"))
tot2=data.table(read.xport("./Original Data/DR2TOT_I.XPT.txt"))
b_pres=data.table(read.xport("./Original Data/BPX_I.XPT.txt"))
ldl=data.table(read.xport("./Original Data/TRIGLY_I.XPT.txt"))
measure=data.table(read.xport("./Original Data/BMX_I.XPT.txt"))
## For each dataset, choose the proper variables and make
## some transformation.
# For demo dataset, we choose seqn, gender, age and race variables
Demo=demo[,.(seqn=SEQN,gender=as.factor(RIAGENDR),
age=RIDAGEYR,race=as.factor(RIDRETH3))]
# For dietary data, we choose seqn, intake fat, intake cholesterol
# for each day.
TOT1=tot1[,.(seqn=SEQN,intake_fat1=DR1TTFAT,
intake_chol1=DR1TCHOL)]
TOT2=tot2[,.(seqn=SEQN,intake_fat2=DR2TTFAT,
intake_chol2=DR2TCHOL)]
## Next we will use the average intake of the two days into
## our model. The average step is as following:
intake_type=c('intake_fat1','intake_fat2','intake_chol1','intake_chol2')
TOT=TOT1%>%merge(.,TOT2,by='seqn',all=FALSE)
TOT=melt(TOT,measure=intake_type)[
,.(seqn,type=factor(variable,intake_type,c(rep('intake_fat',times=2),
rep('intake_chol',times=2))),
variable,value)
][
,.(intake=mean(value,na.rm=TRUE)),by=.(seqn,type)
]
TOT=dcast(TOT,...~type,value.var=c('intake'))
# For blood pressure, we choose seqn, systolic pressures and diastolic
# pressures. We then use the average pressure as the final pressure.
pres_type=c(paste('sys',1:4,sep=''),paste('dia',1:4,sep=''))
B_pres=b_pres[,.(seqn=SEQN,sys1=BPXSY1,sys2=BPXSY2,sys3=BPXSY3,sys4=BPXSY4,
dia1=BPXDI1,dia2=BPXDI2,dia3=BPXDI3,dia4=BPXDI4)]
B_pres=melt(B_pres,measure=pres_type)[,
.(seqn,type=factor(variable,pres_type,
c(rep('s',times=4),rep('d',times=4))),
variable,pressure=value)
][
,.(pres=mean(pressure,na.rm=TRUE)),by=.(seqn,type)
]
B_pres=dcast(B_pres,...~type,value.var=c('pres'))[
,.(seqn,systolic=s,diastolic=d)
]
# For ldl dataset, we choose seqn, LDL-cholesterol and Triglyceride
# for mg/dL.
LDL=ldl[,.(seqn=SEQN,ldl=LBDLDL,triglycerides=LBXTR)]
# For body measure dataset, we choose weight height and bmi as our
# variables.
Measure=measure[,.(seqn=SEQN,weight=BMXWT,height=BMXHT,bmi=BMXBMI)]
## Now merge the datasets into one whole, with the seqn as
## the merging label. By the way, some seqn labels
## should be removed, for they are not included in LDL dataset.
Data=Demo%>%merge(.,TOT,by='seqn',all=FALSE)%>%
merge(.,B_pres,by='seqn',all=FALSE)%>%
merge(.,LDL,by='seqn',all=FALSE)%>%
merge(.,Measure,by='seqn',all=FALSE)%>%
na.omit()
### Using this file for regression: ---------------------------------------------------
## Libraries: -------------------------------------------------------------------------
library(lme4)
library(MASS)
library(car)
## 80: --------------------------------------------------------------------------------
## Remove the seqn variable, and set gender and race as factor variables
DT=Data[,.(gender=as.factor(gender),age,race=as.factor(race),intake_fat,intake_chol,
systolic,diastolic,ldl,triglycerides,weight,height,bmi)]
## First of all, we will fit the model with all variables, and then give
## the residual plot of the model.
L1=lm(ldl~.,data=DT)
summary(L1)
##
## Call:
## lm(formula = ldl ~ ., data = DT)
##
## Residuals:
## Min 1Q Median 3Q Max
## -113.646 -22.757 -2.451 19.297 149.850
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 101.571467 44.874374 2.263 0.0237 *
## gender2 1.171152 1.847729 0.634 0.5262
## age 0.175782 0.038684 4.544 5.78e-06 ***
## race2 3.952555 2.414177 1.637 0.1017
## race3 2.036991 2.093998 0.973 0.3308
## race4 3.506045 2.285949 1.534 0.1252
## race6 5.104577 2.738375 1.864 0.0624 .
## race7 6.036927 3.880265 1.556 0.1199
## intake_fat 0.005288 0.022726 0.233 0.8160
## intake_chol -0.001265 0.004397 -0.288 0.7736
## systolic -0.005915 0.045972 -0.129 0.8976
## diastolic 0.401837 0.057452 6.994 3.41e-12 ***
## triglycerides 0.142759 0.011379 12.546 < 2e-16 ***
## weight 0.301651 0.266228 1.133 0.2573
## height -0.318111 0.269684 -1.180 0.2383
## bmi -0.595490 0.741391 -0.803 0.4219
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 33.34 on 2487 degrees of freedom
## Multiple R-squared: 0.1342, Adjusted R-squared: 0.129
## F-statistic: 25.69 on 15 and 2487 DF, p-value: < 2.2e-16
## Here we give the residual plot of the model
plot(L1$fitted.values,L1$residuals)
## Here, it seems that some transformations should be used upon ldl. Here
## we do the Box-Cox test.
boxcox(L1,plotit=TRUE,lambda=seq(0,1,1/100))
## Here it seems that lambda=0.5 is the best choice, that is, to use sqrt(ldl).
## Here we make the transformation and then do the regression again.
L2=lm(sqrt(ldl)~.,data=DT)
summary(L2)
##
## Call:
## lm(formula = sqrt(ldl) ~ ., data = DT)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.1905 -1.0601 0.0021 1.0178 5.5148
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.005e+01 2.158e+00 4.658 3.36e-06 ***
## gender2 5.965e-02 8.886e-02 0.671 0.5021
## age 8.429e-03 1.860e-03 4.531 6.15e-06 ***
## race2 1.830e-01 1.161e-01 1.576 0.1151
## race3 8.270e-02 1.007e-01 0.821 0.4116
## race4 1.305e-01 1.099e-01 1.187 0.2354
## race6 2.302e-01 1.317e-01 1.748 0.0806 .
## race7 2.744e-01 1.866e-01 1.470 0.1416
## intake_fat 4.259e-04 1.093e-03 0.390 0.6968
## intake_chol -8.613e-05 2.115e-04 -0.407 0.6838
## systolic -3.532e-04 2.211e-03 -0.160 0.8731
## diastolic 2.006e-02 2.763e-03 7.261 5.11e-13 ***
## triglycerides 6.538e-03 5.472e-04 11.948 < 2e-16 ***
## weight 1.492e-02 1.280e-02 1.165 0.2440
## height -1.644e-02 1.297e-02 -1.268 0.2050
## bmi -2.700e-02 3.566e-02 -0.757 0.4489
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.604 on 2487 degrees of freedom
## Multiple R-squared: 0.1327, Adjusted R-squared: 0.1275
## F-statistic: 25.38 on 15 and 2487 DF, p-value: < 2.2e-16
## There are too many variables in the regression model, so here we will do
## the model selection and choose the variables. Here we do both the forward
## and backward selections.
L3=step(L2,direction='both',trace=FALSE)
summary(L3)
##
## Call:
## lm(formula = sqrt(ldl) ~ age + diastolic + triglycerides + weight +
## height, data = DT)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.2425 -1.0646 0.0056 1.0406 5.5286
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.8488834 0.5771649 15.332 < 2e-16 ***
## age 0.0079968 0.0016065 4.978 6.87e-07 ***
## diastolic 0.0203065 0.0025879 7.847 6.28e-15 ***
## triglycerides 0.0064965 0.0005279 12.306 < 2e-16 ***
## weight 0.0049061 0.0017000 2.886 0.00394 **
## height -0.0083689 0.0036375 -2.301 0.02149 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.603 on 2497 degrees of freedom
## Multiple R-squared: 0.1304, Adjusted R-squared: 0.1286
## F-statistic: 74.87 on 5 and 2497 DF, p-value: < 2.2e-16
From the output result, we find out that variables age, diastolic, triglycerides, weight, height are selected, and they are all significant under t-test.
In this model, variables age, diastolic, triglycerides and weight are positive correlated to the fitted level of ldl, while height is negative correlated: with other variables fixed, one year of age increase leads to 0.008 unit increase in \(\sqrt{ldl}\), and 1 unit diastolic increase leads to 0.02 unit increase in \(\sqrt{ldl}\); 1 unit increase in triglycerides leads to 0.006 unit increase in \(\sqrt{ldl}\), and for weight this will lead to 0.005 unit increase in \(\sqrt{ldl}\); for height, this will lead to 0.008 unit decrease in \(\sqrt{ldl}\). The \(R^2\) is 0.1304, and the residual standard error is 1.603.
## By the way, in the models before, we didn't consider transformations
## upon predictors; in the coming part, we will consider adding some
## nonlinear terms.
crPlots(L2,layout=c(4,3))
## From the partial residual plots, we can find out that for triglycerides and age,
## some nonlinear transformation forms should be add. We add this term, and the
## regression result is as following:
L4=lm(sqrt(ldl)~gender+age+race+intake_fat+intake_chol+systolic+diastolic+
weight+height+bmi+triglycerides+I(triglycerides^2)+I(age^2),data=DT)
summary(L4)
##
## Call:
## lm(formula = sqrt(ldl) ~ gender + age + race + intake_fat + intake_chol +
## systolic + diastolic + weight + height + bmi + triglycerides +
## I(triglycerides^2) + I(age^2), data = DT)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.2795 -0.9630 -0.0056 0.9958 5.4639
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.155e+01 2.065e+00 5.593 2.47e-08 ***
## gender2 -7.780e-02 8.572e-02 -0.908 0.36421
## age 1.075e-01 8.753e-03 12.282 < 2e-16 ***
## race2 8.510e-02 1.109e-01 0.767 0.44310
## race3 1.954e-01 9.673e-02 2.020 0.04349 *
## race4 2.324e-01 1.053e-01 2.207 0.02739 *
## race6 1.303e-01 1.258e-01 1.035 0.30054
## race7 3.578e-01 1.782e-01 2.008 0.04472 *
## intake_fat -7.455e-05 1.043e-03 -0.071 0.94303
## intake_chol -1.705e-04 2.019e-04 -0.844 0.39857
## systolic 3.840e-03 2.141e-03 1.794 0.07298 .
## diastolic 6.303e-03 2.858e-03 2.206 0.02750 *
## weight 2.520e-02 1.223e-02 2.060 0.03950 *
## height -3.457e-02 1.246e-02 -2.775 0.00556 **
## bmi -7.122e-02 3.413e-02 -2.087 0.03703 *
## triglycerides 2.150e-02 1.660e-03 12.953 < 2e-16 ***
## I(triglycerides^2) -5.019e-05 5.005e-06 -10.027 < 2e-16 ***
## I(age^2) -1.138e-03 9.593e-05 -11.862 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.53 on 2485 degrees of freedom
## Multiple R-squared: 0.2114, Adjusted R-squared: 0.206
## F-statistic: 39.2 on 17 and 2485 DF, p-value: < 2.2e-16
## Just the same, do the model selection.
L5=step(L4,direction='both',trace=FALSE)
summary(L5)
##
## Call:
## lm(formula = sqrt(ldl) ~ age + systolic + diastolic + weight +
## height + bmi + triglycerides + I(triglycerides^2) + I(age^2),
## data = DT)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.2347 -0.9663 -0.0208 1.0166 5.4756
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.109e+01 2.011e+00 5.516 3.82e-08 ***
## age 1.047e-01 8.550e-03 12.249 < 2e-16 ***
## systolic 4.101e-03 2.091e-03 1.961 0.0500 *
## diastolic 6.455e-03 2.832e-03 2.279 0.0228 *
## weight 2.636e-02 1.221e-02 2.159 0.0310 *
## height -3.121e-02 1.214e-02 -2.571 0.0102 *
## bmi -7.503e-02 3.398e-02 -2.208 0.0274 *
## triglycerides 2.116e-02 1.620e-03 13.060 < 2e-16 ***
## I(triglycerides^2) -4.954e-05 4.948e-06 -10.011 < 2e-16 ***
## I(age^2) -1.105e-03 9.325e-05 -11.849 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.53 on 2493 degrees of freedom
## Multiple R-squared: 0.2085, Adjusted R-squared: 0.2057
## F-statistic: 72.98 on 9 and 2493 DF, p-value: < 2.2e-16
After model selection, we can find out that terms age, systolic, diastolic, weight, height, bmi, triglycerides, \(triglycerides^2\) and \(age^2\) are selected, and they are significant under t-test. The height bmi and the two square terms are negatively correlated with ldl, with other variables positively correlated with ldl. The residual standard error changes to 1.53, and the \(R^2\) increases to 0.2085, which means that this model performs better than the one without square terms.
* import demographic data
import sasxport "./Original Data/DEMO_I.XPT.txt", clear
* rename variables
rename riagendr gender
rename ridageyr age
rename ridreth3 race
* select variables of focus
keep seqn gender age race
* save the cleaned demographics data
save "./Xiru Lyu/Data/demo.dta", replace
* import diet (day 1) data
import sasxport "./Original Data/DR1TOT_I.XPT.txt", clear
* rename variables
rename dr1ttfat fat1
rename dr1tchol chol1
* select variables of interest
keep seqn fat1 chol1
* save the cleaned diet (day 1) dataset
save "./Xiru Lyu/Data/diet1.dta", replace
* import diet (day 2) data
import sasxport "./Original Data/DR2TOT_I.XPT.txt", clear
* rename variables
rename dr2ttfat fat2
rename dr2tchol chol2
* select variables of focus
keep seqn fat2 chol2
* save the cleaned diet (day 2) dataset
save "./Xiru Lyu/Data/diet2.dta", replace
* import LDL & triglyceride data
import sasxport "./Original Data/TRIGLY_I.XPT.txt", clear
* rename variables
rename lbdldl ldl
rename lbxtr triglyceride
* select variables of interest
keep seqn ldl trig
* save the cleaned cholesterol dataset
save "./Xiru Lyu/Data/ldl.dta", replace
* import blood pressure data
import sasxport "./Original Data/BPX_I.XPT.txt", clear
* rename variables
rename bpxsy1 sy1
rename bpxsy2 sy2
rename bpxsy3 sy3
rename bpxsy4 sy4
rename bpxdi1 di1
rename bpxdi2 di2
rename bpxdi3 di3
rename bpxdi4 di4
* compute averaged systolic and diastolic blood pressure for each participant
egen systolic = rowmean(sy1 sy2 sy3 sy4)
egen diastolic = rowmean(di1 di2 di3 di4)
* select variables of interest
keep seqn systolic diastolic
* save the cleaned blood pressure dataset
save "./Xiru Lyu/Data/bp.dta", replace
* import body measure data
import sasxport "./Original Data/BMX_I.XPT.txt", clear
* rename variables
rename bmxwt weight
rename bmxbmi bmi
rename bmxht height
* select variables of interest
keep seqn weight height bmi
* merge datasets by seqn
merge 1:1 seqn using "./Xiru Lyu/Data/demo.dta"
keep if _merge == 3
drop _merge
merge 1:1 seqn using "./Xiru Lyu/Data/diet1.dta"
keep if _merge == 3
drop _merge
merge 1:1 seqn using "./Xiru Lyu/Data/diet2.dta"
keep if _merge == 3
drop _merge
merge 1:1 seqn using "./Xiru Lyu/Data/bp.dta"
keep if _merge == 3
drop _merge
merge 1:1 seqn using "./Xiru Lyu/Data/ldl.dta"
keep if _merge == 3
drop _merge
* compute averaged intakes of fat and cholesterol
egen fat = rowmean(fat1 fat2)
egen chol = rowmean(chol1 chol2)
* drop extra columns
drop fat1 fat2 chol1 chol2
* drop rows with missing values
foreach var of varlist age bmi chol diastolic fat gender height ldl race ///
seqn systolic triglyceride weight{
drop if missing(`var')
}
* save the dataset for data analysis
save "./Xiru Lyu/Data/final.dta", replace
* transform the dependent variable
generate ldl2 = sqrt(ldl)
* fit a multiple linear regression model
regress ldl2 age i.race i.gender bmi weight height diastolic systolic chol ///
fat trig
Regression Result of Full Model (2)
I first fitted the full model, including all possible covariates and the transformed dependent variable, and then I used forward and backward stepwise selections for model selections. To be consistent with the result produced by R, I used the result from the backward selection for later AIC/BIC comparison.
* backward selection
xi: stepwise, pr(.1): regress ldl2 age i.race i.gender bmi weight height ///
diastolic systolic chol fat triglyceride
Backward Selection Result of Full Model (2)
* forward selection
xi: stepwise, pe(.1): regress ldl2 age i.race i.gender bmi weight height ///
diastolic systolioc chol fat triglyceride
Forward Selection Result of Full Model (2)
* transform covariates
generate triglyceride2 = triglyceride^2
generate age2 = age^2
* fit a multiple linear regression model
regress ldl2 age age2 i.race i.gender bmi weight height diastolic systolic ///
chol fat triglyceride triglyceride2
I then fitted another model that contains two extra transformed independent variables. With the stepwise model selection procedure, I kept the model selected by the backward selection as the one for further AIC/BIC comparison so that my result is consistent with the result produced by R.
Regression Result of Full Model (3)
* backward selection
xi: stepwise, pr(.05): regress ldl2 age age2 i.race i.gender bmi weight ///
height diastolic systolic chol fat triglyceride triglyceride2
Backward Selection Result of Full Model (3)
* forward selection
xi: stepwise, pe(.05): regress ldl2 age age2 i.race i.gender bmi weight ///
height diastolic systolic chol fat triglyceride triglyceride2
Forward Selection Result of Full Model (3)
* compare AIC & BIC of two nested models
regress ldl2 age height weight diastolic triglyceride
estat ic
AIC for Model 2_a
regress ldl2 age age2 height weight bmi diastolic systolic triglyceride ///
triglyceride2
estat ic
AIC for Model 2_b
A comparison for AIC/BIC for two nested models shows that the model \(\mathbf{\sqrt{LDL}}\) ~ \(\mathbf{age + age^2 + height + weight + BMI + triglyceride + triglyceride^2 + diastolic + systolic}\) is a better one. Inferential statistics for the model is produced below. Covariates age, triglyceride, weight, diastolic and systolic blood pressures are positively correlated with the level of LDL, while {age2}, {triglyceride2} and BMI are negatively correlated with the response variable.
Specifically, with other variables fixed, one year increase in age leads to approximately 0.10 unit of increase in \(\sqrt{ldl}\). Also, the rate of increase for the level of \(\sqrt{ldl}\) slows down as one ages. One unit of increase in diastolic blood pressure can increase \(\sqrt{ldl}\) by 0.006 unit. One unit of increase in systolic blood pressure can increase \(\sqrt{ldl}\) by 0.004 unit. One unit of increase in BMI decreases \(\sqrt{ldl}\) by 0.075 unit. One unit of increase in weight and triglyceride can bump up \(\sqrt{ldl}\) by .026 and .021 unit, respectively. Finally, one unit increase in height leads to approximately 0.026 unit of decrease in \(\sqrt{ldl}\).
import pandas as pd
import numpy as np
LDLdata=pd.read_sas(r'./Original Data/TRIGLY_I.XPT',encoding='utf8')
DR1=pd.read_sas(r'./Original Data/DR1TOT_I.XPT',encoding='utf8')
DR2=pd.read_sas(r'./Original Data/DR2TOT_I.XPT',encoding='utf8')
BPX=pd.read_sas(r'./Original Data/BPX_I.XPT',encoding='utf8')
DEMO=pd.read_sas(r'./Original Data/DEMO_I.XPT',encoding='utf8')
BMI=pd.read_sas(r'./Original Data/BMX_I.XPT',encoding='utf8')
LDL=LDLdata[['SEQN','LBXTR', 'LBDLDL']]
#WTSAF2YR:MEC weight
#LBXTR: triglyceride(mg/dl)
#LBDLDL: LDL mg/dl
BloodP=BPX[['SEQN']]
BloodP['BPXSY']=np.nanmean(BPX[['BPXSY1','BPXSY2','BPXSY3','BPXSY4']],axis=1)
## /Users/lihuayu/anaconda3/bin/python:1: RuntimeWarning: Mean of empty slice
## /Users/lihuayu/anaconda3/bin/python:1: SettingWithCopyWarning:
## A value is trying to be set on a copy of a slice from a DataFrame.
## Try using .loc[row_indexer,col_indexer] = value instead
##
## See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
BloodP['BPXDI']=np.nanmean(BPX[['BPXDI1','BPXDI2','BPXDI3','BPXDI4']],axis=1)
BloodP
## SEQN BPXSY BPXDI
## 0 83732.0 122.666667 65.333333
## 1 83733.0 140.000000 86.000000
## 2 83734.0 135.333333 45.333333
## 3 83735.0 134.000000 70.000000
## 4 83736.0 104.000000 60.000000
## 5 83737.0 119.333333 58.666667
## 6 83738.0 100.000000 48.000000
## 7 83739.0 NaN NaN
## 8 83740.0 NaN NaN
## 9 83741.0 111.333333 72.666667
## 10 83742.0 118.000000 70.666667
## 11 83743.0 NaN NaN
## 12 83744.0 179.333333 111.333333
## 13 83745.0 111.333333 78.000000
## 14 83746.0 NaN NaN
## 15 83747.0 148.000000 92.000000
## 16 83748.0 NaN NaN
## 17 83749.0 113.333333 58.666667
## 18 83750.0 110.666667 72.000000
## 19 83751.0 103.333333 53.333333
## 20 83752.0 104.000000 50.000000
## 21 83753.0 121.333333 61.333333
## 22 83754.0 118.666667 72.000000
## 23 83755.0 133.333333 81.333333
## 24 83756.0 120.000000 64.000000
## 25 83757.0 142.666667 62.666667
## 26 83759.0 104.000000 68.666667
## 27 83760.0 NaN NaN
## 28 83761.0 107.333333 61.333333
## 29 83762.0 140.000000 76.666667
## ... ... ... ...
## 9514 93672.0 NaN NaN
## 9515 93673.0 NaN NaN
## 9516 93674.0 94.666667 56.000000
## 9517 93675.0 112.666667 59.333333
## 9518 93676.0 115.333333 76.666667
## 9519 93677.0 113.333333 75.333333
## 9520 93678.0 NaN NaN
## 9521 93679.0 136.000000 68.666667
## 9522 93680.0 112.000000 58.666667
## 9523 93682.0 126.000000 81.333333
## 9524 93683.0 NaN NaN
## 9525 93684.0 110.666667 71.333333
## 9526 93685.0 127.333333 55.333333
## 9527 93686.0 112.000000 70.000000
## 9528 93687.0 114.666667 61.333333
## 9529 93688.0 NaN NaN
## 9530 93689.0 164.000000 66.000000
## 9531 93690.0 115.333333 62.000000
## 9532 93691.0 112.000000 76.000000
## 9533 93692.0 NaN NaN
## 9534 93693.0 NaN NaN
## 9535 93694.0 NaN NaN
## 9536 93695.0 111.333333 47.333333
## 9537 93696.0 116.000000 72.000000
## 9538 93697.0 148.000000 55.333333
## 9539 93698.0 NaN NaN
## 9540 93699.0 NaN NaN
## 9541 93700.0 104.666667 65.333333
## 9542 93701.0 114.000000 48.666667
## 9543 93702.0 118.666667 66.000000
##
## [9544 rows x 3 columns]
drday1=DR1[['SEQN','DR1TTFAT','DR1TCHOL']]
drday2=DR2[['SEQN','DR2TTFAT','DR2TCHOL']]
drboth=pd.merge(drday1,drday2,how='inner',on='SEQN')
drboth['FAT']=np.nanmean(drboth[['DR1TTFAT','DR2TTFAT']],axis=1)
drboth['CHOL']=np.nanmean(drboth[['DR1TCHOL','DR2TCHOL']],axis=1)
dr=drboth[['SEQN','FAT','CHOL']]
demo=DEMO[['SEQN','RIAGENDR','RIDAGEYR','RIDRETH3']]
#ID,GENDER,AGE,RACE(FACTOR)
demo.rename(columns={'RIAGENDR':'GENDER','RIDAGEYR':'AGE','RIDRETH3':'RACE'},inplace=True)
## /Users/lihuayu/anaconda3/lib/python3.7/site-packages/pandas/core/frame.py:4025: SettingWithCopyWarning:
## A value is trying to be set on a copy of a slice from a DataFrame
##
## See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
## return super(DataFrame, self).rename(**kwargs)
BMI=BMI[['SEQN','BMXHT','BMXWT','BMXBMI']]
#ID,HEIGHT,WEIGHT
BMI.columns=['SEQN','HEIGHT','WEIGHT','BMI']
Data=pd.merge(LDL,BloodP,how='inner',on='SEQN')
Data=pd.merge(Data,dr,how='inner',on='SEQN')
Data=pd.merge(Data,demo,how='inner',on='SEQN')
Data=pd.merge(Data,BMI,how='inner',on='SEQN')
Data=Data.dropna(axis=0)
Data.columns
## Index(['SEQN', 'LBXTR', 'LBDLDL', 'BPXSY', 'BPXDI', 'FAT', 'CHOL', 'GENDER',
## 'AGE', 'RACE', 'HEIGHT', 'WEIGHT', 'BMI'],
## dtype='object')
Data.to_csv(r'./kaliu python/Data1.csv',index=None)
import pandas as pd
import numpy as np
from sklearn import linear_model
from scipy import stats
import statsmodels.formula.api as smf
import matplotlib.pyplot as plt
import pylab
import statsmodels.api as sm
## code from github, link https://github.com/talhahascelik/python_stepwiseSelection/blob/master/stepwiseSelection.py
#Copyright 2019 Sinan Talha Hascelik
#
#Licensed under the Apache License, Version 2.0 (the "License");
#you may not use this file except in compliance with the License.
#You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
#Unless required by applicable law or agreed to in writing, software
#distributed under the License is distributed on an "AS IS" BASIS,
#WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
#See the License for the specific language governing permissions and
#limitations under the License.
def forwardSelection(X, y, model_type ="linear",elimination_criteria = "aic", varchar_process = "dummy_dropfirst", sl=0.05):
"""
Forward Selection is a function, based on regression models, that returns significant features and selection iterations.\n
Required Libraries: pandas, numpy, statmodels
Parameters
----------
X : Independent variables (Pandas Dataframe)\n
y : Dependent variable (Pandas Series, Pandas Dataframe)\n
model_type : 'linear' or 'logistic'\n
elimination_criteria : 'aic', 'bic', 'r2', 'adjr2' or None\n
'aic' refers Akaike information criterion\n
'bic' refers Bayesian information criterion\n
'r2' refers R-squared (Only works on linear model type)\n
'r2' refers Adjusted R-squared (Only works on linear model type)\n
varchar_process : 'drop', 'dummy' or 'dummy_dropfirst'\n
'drop' drops varchar features\n
'dummy' creates dummies for all levels of all varchars\n
'dummy_dropfirst' creates dummies for all levels of all varchars, and drops first levels\n
sl : Significance Level (default: 0.05)\n
Returns
-------
columns(list), iteration_logs(str)\n\n
Not Returns a Model
Tested On
---------
Python v3.6.7, Pandas v0.23.4, Numpy v1.15.04, StatModels v0.9.0
See Also
--------
https://en.wikipedia.org/wiki/Stepwise_regression
"""
X = __varcharProcessing__(X,varchar_process = varchar_process)
return __forwardSelectionRaw__(X, y, model_type = model_type,elimination_criteria = elimination_criteria , sl=sl)
def backwardSelection(X, y, model_type ="linear",elimination_criteria = "aic", varchar_process = "dummy_dropfirst", sl=0.05):
"""
Backward Selection is a function, based on regression models, that returns significant features and selection iterations.\n
Required Libraries: pandas, numpy, statmodels
Parameters
----------
X : Independent variables (Pandas Dataframe)\n
y : Dependent variable (Pandas Series, Pandas Dataframe)\n
model_type : 'linear' or 'logistic'\n
elimination_criteria : 'aic', 'bic', 'r2', 'adjr2' or None\n
'aic' refers Akaike information criterion\n
'bic' refers Bayesian information criterion\n
'r2' refers R-squared (Only works on linear model type)\n
'r2' refers Adjusted R-squared (Only works on linear model type)\n
varchar_process : 'drop', 'dummy' or 'dummy_dropfirst'\n
'drop' drops varchar features\n
'dummy' creates dummies for all levels of all varchars\n
'dummy_dropfirst' creates dummies for all levels of all varchars, and drops first levels\n
sl : Significance Level (default: 0.05)\n
Returns
-------
columns(list), iteration_logs(str)\n\n
Not Returns a Model
Tested On
---------
Python v3.6.7, Pandas v0.23.4, Numpy v1.15.04, StatModels v0.9.0
See Also
--------
https://en.wikipedia.org/wiki/Stepwise_regression
"""
X = __varcharProcessing__(X,varchar_process = varchar_process)
return __backwardSelectionRaw__(X, y, model_type = model_type,elimination_criteria = elimination_criteria , sl=sl)
def __varcharProcessing__(X, varchar_process = "dummy_dropfirst"):
dtypes = X.dtypes
if varchar_process == "drop":
X = X.drop(columns = dtypes[dtypes == np.object].index.tolist())
print("Character Variables (Dropped):", dtypes[dtypes == np.object].index.tolist())
elif varchar_process == "dummy":
X = pd.get_dummies(X,drop_first=False)
print("Character Variables (Dummies Generated):", dtypes[dtypes == np.object].index.tolist())
elif varchar_process == "dummy_dropfirst":
X = pd.get_dummies(X,drop_first=True)
print("Character Variables (Dummies Generated, First Dummies Dropped):", dtypes[dtypes == np.object].index.tolist())
else:
X = pd.get_dummies(X,drop_first=True)
print("Character Variables (Dummies Generated, First Dummies Dropped):", dtypes[dtypes == np.object].index.tolist())
X["intercept"] = 1
cols = X.columns.tolist()
cols = cols[-1:] + cols[:-1]
X = X[cols]
return X
def __forwardSelectionRaw__(X, y, model_type ="linear",elimination_criteria = "aic", sl=0.05):
iterations_log = ""
cols = X.columns.tolist()
def regressor(y,X, model_type=model_type):
if model_type == "linear":
regressor = sm.OLS(y, X).fit()
elif model_type == "logistic":
regressor = sm.Logit(y, X).fit()
else:
print("\nWrong Model Type : "+ model_type +"\nLinear model type is seleted.")
model_type = "linear"
regressor = sm.OLS(y, X).fit()
return regressor
selected_cols = ["intercept"]
other_cols = cols.copy()
other_cols.remove("intercept")
model = regressor(y, X[selected_cols])
if elimination_criteria == "aic":
criteria = model.aic
elif elimination_criteria == "bic":
criteria = model.bic
elif elimination_criteria == "r2" and model_type =="linear":
criteria = model.rsquared
elif elimination_criteria == "adjr2" and model_type =="linear":
criteria = model.rsquared_adj
for i in range(X.shape[1]):
pvals = pd.DataFrame(columns = ["Cols","Pval"])
for j in other_cols:
model = regressor(y, X[selected_cols+[j]])
pvals = pvals.append(pd.DataFrame([[j, model.pvalues[j]]],columns = ["Cols","Pval"]),ignore_index=True)
pvals = pvals.sort_values(by = ["Pval"]).reset_index(drop=True)
pvals = pvals[pvals.Pval<=sl]
if pvals.shape[0] > 0:
model = regressor(y, X[selected_cols+[pvals["Cols"][0]]])
iterations_log += str("\nEntered : "+pvals["Cols"][0] + "\n")
iterations_log += "\n\n"+str(model.summary())+"\nAIC: "+ str(model.aic) + "\nBIC: "+ str(model.bic)+"\n\n"
if elimination_criteria == "aic":
new_criteria = model.aic
if new_criteria < criteria:
print("Entered :", pvals["Cols"][0], "\tAIC :", model.aic)
selected_cols.append(pvals["Cols"][0])
other_cols.remove(pvals["Cols"][0])
criteria = new_criteria
else:
print("break : Criteria")
break
elif elimination_criteria == "bic":
new_criteria = model.bic
if new_criteria < criteria:
print("Entered :", pvals["Cols"][0], "\tBIC :", model.bic)
selected_cols.append(pvals["Cols"][0])
other_cols.remove(pvals["Cols"][0])
criteria = new_criteria
else:
print("break : Criteria")
break
elif elimination_criteria == "r2" and model_type =="linear":
new_criteria = model.rsquared
if new_criteria > criteria:
print("Entered :", pvals["Cols"][0], "\tR2 :", model.rsquared)
selected_cols.append(pvals["Cols"][0])
other_cols.remove(pvals["Cols"][0])
criteria = new_criteria
else:
print("break : Criteria")
break
elif elimination_criteria == "adjr2" and model_type =="linear":
new_criteria = model.rsquared_adj
if new_criteria > criteria:
print("Entered :", pvals["Cols"][0], "\tAdjR2 :", model.rsquared_adj)
selected_cols.append(pvals["Cols"][0])
other_cols.remove(pvals["Cols"][0])
criteria = new_criteria
else:
print("Break : Criteria")
break
else:
print("Entered :", pvals["Cols"][0])
selected_cols.append(pvals["Cols"][0])
other_cols.remove(pvals["Cols"][0])
else:
print("Break : Significance Level")
break
model = regressor(y, X[selected_cols])
if elimination_criteria == "aic":
criteria = model.aic
elif elimination_criteria == "bic":
criteria = model.bic
elif elimination_criteria == "r2" and model_type =="linear":
criteria = model.rsquared
elif elimination_criteria == "adjr2" and model_type =="linear":
criteria = model.rsquared_adj
print(model.summary())
print("AIC: "+str(model.aic))
print("BIC: "+str(model.bic))
print("Final Variables:", selected_cols)
return selected_cols, iterations_log
def __backwardSelectionRaw__(X, y, model_type ="linear",elimination_criteria = "aic", sl=0.05):
iterations_log = ""
last_eleminated = ""
cols = X.columns.tolist()
def regressor(y,X, model_type=model_type):
if model_type =="linear":
regressor = sm.OLS(y, X).fit()
elif model_type == "logistic":
regressor = sm.Logit(y, X).fit()
else:
print("\nWrong Model Type : "+ model_type +"\nLinear model type is seleted.")
model_type = "linear"
regressor = sm.OLS(y, X).fit()
return regressor
for i in range(X.shape[1]):
if i != 0 :
if elimination_criteria == "aic":
criteria = model.aic
new_model = regressor(y,X)
new_criteria = new_model.aic
if criteria < new_criteria:
print("Regained : ", last_eleminated)
iterations_log += "\n"+str(new_model.summary())+"\nAIC: "+ str(new_model.aic) + "\nBIC: "+ str(new_model.bic)+"\n"
iterations_log += str("\n\nRegained : "+last_eleminated + "\n\n")
break
elif elimination_criteria == "bic":
criteria = model.bic
new_model = regressor(y,X)
new_criteria = new_model.bic
if criteria < new_criteria:
print("Regained : ", last_eleminated)
iterations_log += "\n"+str(new_model.summary())+"\nAIC: "+ str(new_model.aic) + "\nBIC: "+ str(new_model.bic)+"\n"
iterations_log += str("\n\nRegained : "+last_eleminated + "\n\n")
break
elif elimination_criteria == "adjr2" and model_type =="linear":
criteria = model.rsquared_adj
new_model = regressor(y,X)
new_criteria = new_model.rsquared_adj
if criteria > new_criteria:
print("Regained : ", last_eleminated)
iterations_log += "\n"+str(new_model.summary())+"\nAIC: "+ str(new_model.aic) + "\nBIC: "+ str(new_model.bic)+"\n"
iterations_log += str("\n\nRegained : "+last_eleminated + "\n\n")
break
elif elimination_criteria == "r2" and model_type =="linear":
criteria = model.rsquared
new_model = regressor(y,X)
new_criteria = new_model.rsquared
if criteria > new_criteria:
print("Regained : ", last_eleminated)
iterations_log += "\n"+str(new_model.summary())+"\nAIC: "+ str(new_model.aic) + "\nBIC: "+ str(new_model.bic)+"\n"
iterations_log += str("\n\nRegained : "+last_eleminated + "\n\n")
break
else:
new_model = regressor(y,X)
model = new_model
iterations_log += "\n"+str(model.summary())+"\nAIC: "+ str(model.aic) + "\nBIC: "+ str(model.bic)+"\n"
else:
model = regressor(y,X)
iterations_log += "\n"+str(model.summary())+"\nAIC: "+ str(model.aic) + "\nBIC: "+ str(model.bic)+"\n"
maxPval = max(model.pvalues)
cols = X.columns.tolist()
if maxPval > sl:
for j in cols:
if (model.pvalues[j] == maxPval):
print("Eliminated :" ,j)
iterations_log += str("\n\nEliminated : "+j+ "\n\n")
del X[j]
last_eleminated = j
else:
break
print(str(model.summary())+"\nAIC: "+ str(model.aic) + "\nBIC: "+ str(model.bic))
print("Final Variables:", cols)
iterations_log += "\n"+str(model.summary())+"\nAIC: "+ str(model.aic) + "\nBIC: "+ str(model.bic)+"\n"
return cols, iterations_log
Data=pd.read_csv(r'./kaliu python/Data1.csv')
Data=Data.drop(['SEQN'],axis=1)#drop the id variable 'SEQN'
RACE=pd.get_dummies(Data['RACE'])
RACENAME=['Mexican American','Other Hispanic','Non-Hispanic White','Non-Hispanic Black','Non-Hispanic Asian','Other Race']
RACE.columns=[ i for i in RACENAME]
RACE=RACE.drop(['Mexican American'],axis=1)
GENDER=pd.get_dummies(Data['GENDER'])
gendername=['Male','Female']
GENDER.columns=[i for i in gendername]
GENDER=GENDER.drop(['Male'],axis=1)
Data=Data.drop(['GENDER','RACE'],axis=1)
Data=pd.concat([Data,GENDER,RACE],axis=1)
y1=Data[['LBDLDL']] ## lm reg, the response= LBDLDL (LDL)
x1=Data.drop(['LBDLDL'],axis=1) # others except LDL are the variables
x1=sm.add_constant(x1) ## add intercept, for python, add 1 colunms by hand
## /Users/lihuayu/anaconda3/lib/python3.7/site-packages/numpy/core/fromnumeric.py:2495: FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.
## return ptp(axis=axis, out=out, **kwargs)
lm1=sm.OLS(y1.astype(float),x1.astype(float)).fit() # OLS y on X+1
lm1.summary() # coefficient table
## <class 'statsmodels.iolib.summary.Summary'>
## """
## OLS Regression Results
## ==============================================================================
## Dep. Variable: LBDLDL R-squared: 0.134
## Model: OLS Adj. R-squared: 0.129
## Method: Least Squares F-statistic: 25.69
## Date: Wed, 11 Dec 2019 Prob (F-statistic): 2.47e-67
## Time: 23:24:47 Log-Likelihood: -12321.
## No. Observations: 2503 AIC: 2.467e+04
## Df Residuals: 2487 BIC: 2.477e+04
## Df Model: 15
## Covariance Type: nonrobust
## ======================================================================================
## coef std err t P>|t| [0.025 0.975]
## --------------------------------------------------------------------------------------
## const 101.5715 44.874 2.263 0.024 13.576 189.566
## LBXTR 0.1428 0.011 12.546 0.000 0.120 0.165
## BPXSY -0.0059 0.046 -0.129 0.898 -0.096 0.084
## BPXDI 0.4018 0.057 6.994 0.000 0.289 0.514
## FAT 0.0053 0.023 0.233 0.816 -0.039 0.050
## CHOL -0.0013 0.004 -0.288 0.774 -0.010 0.007
## AGE 0.1758 0.039 4.544 0.000 0.100 0.252
## HEIGHT -0.3181 0.270 -1.180 0.238 -0.847 0.211
## WEIGHT 0.3017 0.266 1.133 0.257 -0.220 0.824
## BMI -0.5955 0.741 -0.803 0.422 -2.049 0.858
## Female 1.1712 1.848 0.634 0.526 -2.452 4.794
## Other Hispanic 3.9526 2.414 1.637 0.102 -0.781 8.687
## Non-Hispanic White 2.0370 2.094 0.973 0.331 -2.069 6.143
## Non-Hispanic Black 3.5060 2.286 1.534 0.125 -0.977 7.989
## Non-Hispanic Asian 5.1046 2.738 1.864 0.062 -0.265 10.474
## Other Race 6.0369 3.880 1.556 0.120 -1.572 13.646
## ==============================================================================
## Omnibus: 121.293 Durbin-Watson: 1.991
## Prob(Omnibus): 0.000 Jarque-Bera (JB): 161.325
## Skew: 0.469 Prob(JB): 9.30e-36
## Kurtosis: 3.816 Cond. No. 2.87e+04
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## [2] The condition number is large, 2.87e+04. This might indicate that there are
## strong multicollinearity or other numerical problems.
## """
From result, many coefficient are not significant, so to better select variables, we are going to see the resdisual condition and use boxcox to help.
res1=lm1.resid
y1_fit=lm1.predict(x1)
fig1=plt.figure(figsize=(8,6))
plt.plot(y1_fit,res1,'o''c') #'0' means dots,''''means no line between dots, 'c' color
plt.title('Residuals against Y-fited')
plt.ylabel('Residuals')
plt.xlabel('Y-fitted value')
plt.show()
res = lm1.resid # residuals
stats.probplot(res, dist="norm", plot=pylab) # QQplot, simliar to fig = sm.qqplot(res)
## ((array([-3.45329821, -3.20637418, -3.06964752, ..., 3.06964752,
## 3.20637418, 3.45329821]), array([-113.64623725, -108.54537041, -89.63120833, ..., 136.31796236,
## 145.92859063, 149.84990144])), (33.06076150557924, 1.636389036476502e-12, 0.9934850678740555))
pylab.show()
Probability Plot
First, we plot y_fit versus residual, then we plot QQplot of residual, and we can see that the QQplot shows that the residual is not really normal.
#boxcox
y2=np.array(y1).flatten()
fig2=plt.figure()
ax = fig2.add_subplot(111)
prob=stats.boxcox_normplot(y2,0,1,plot=ax)
_, maxlog=stats.boxcox(y2)
ax.axvline(maxlog,color='r')
## <matplotlib.lines.Line2D object at 0x13bf661d0>
plt.show()
From the plot, we can tansform the y value to \(y^\lambda\), when lambda is around 0.4, then result seems best,to make it simple we pick \(\lambda=0.5\).
Data['LDL']=(Data[['LBDLDL']])**0.5 # transform the y value to y**0.5
Data=Data.drop(['LBDLDL'],axis=1)
y3=Data[['LDL']]
x3=Data.drop(['LDL'],axis=1)
final_vars,_=forwardSelection(x3,y3,model_type='linear',elimination_criteria='aic')
## Character Variables (Dummies Generated, First Dummies Dropped): []
## Entered : LBXTR AIC : 9574.218891776083
## Entered : BPXDI AIC : 9503.117731043025
## Entered : AGE AIC : 9475.727581279705
## Entered : BMI AIC : 9470.123755541823
## Break : Significance Level
## OLS Regression Results
## ==============================================================================
## Dep. Variable: LDL R-squared: 0.130
## Model: OLS Adj. R-squared: 0.128
## Method: Least Squares F-statistic: 93.04
## Date: Wed, 11 Dec 2019 Prob (F-statistic): 7.62e-74
## Time: 23:24:49 Log-Likelihood: -4730.1
## No. Observations: 2503 AIC: 9470.
## Df Residuals: 2498 BIC: 9499.
## Df Model: 4
## Covariance Type: nonrobust
## ==============================================================================
## coef std err t P>|t| [0.025 0.975]
## ------------------------------------------------------------------------------
## intercept 7.4982 0.199 37.641 0.000 7.108 7.889
## LBXTR 0.0065 0.001 12.356 0.000 0.005 0.008
## BPXDI 0.0199 0.003 7.769 0.000 0.015 0.025
## AGE 0.0081 0.002 5.007 0.000 0.005 0.011
## BMI 0.0130 0.005 2.757 0.006 0.004 0.022
## ==============================================================================
## Omnibus: 8.993 Durbin-Watson: 1.996
## Prob(Omnibus): 0.011 Jarque-Bera (JB): 11.361
## Skew: -0.017 Prob(JB): 0.00341
## Kurtosis: 3.328 Cond. No. 895.
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## AIC: 9470.123755541823
## BIC: 9499.249981998983
## Final Variables: ['intercept', 'LBXTR', 'BPXDI', 'AGE', 'BMI']
final_vars,_=backwardSelection(x3,y3,model_type='linear',elimination_criteria='aic')
## Character Variables (Dummies Generated, First Dummies Dropped): []
## Eliminated : BPXSY
## Eliminated : FAT
## Eliminated : CHOL
## Eliminated : Female
## Eliminated : BMI
## Eliminated : Non-Hispanic White
## Eliminated : Non-Hispanic Black
## Eliminated : Other Hispanic
## Eliminated : Other Race
## Eliminated : Non-Hispanic Asian
## OLS Regression Results
## ==============================================================================
## Dep. Variable: LDL R-squared: 0.130
## Model: OLS Adj. R-squared: 0.129
## Method: Least Squares F-statistic: 74.87
## Date: Wed, 11 Dec 2019 Prob (F-statistic): 2.88e-73
## Time: 23:24:49 Log-Likelihood: -4729.0
## No. Observations: 2503 AIC: 9470.
## Df Residuals: 2497 BIC: 9505.
## Df Model: 5
## Covariance Type: nonrobust
## ==============================================================================
## coef std err t P>|t| [0.025 0.975]
## ------------------------------------------------------------------------------
## intercept 8.8489 0.577 15.332 0.000 7.717 9.981
## LBXTR 0.0065 0.001 12.306 0.000 0.005 0.008
## BPXDI 0.0203 0.003 7.847 0.000 0.015 0.025
## AGE 0.0080 0.002 4.978 0.000 0.005 0.011
## HEIGHT -0.0084 0.004 -2.301 0.021 -0.016 -0.001
## WEIGHT 0.0049 0.002 2.886 0.004 0.002 0.008
## ==============================================================================
## Omnibus: 8.921 Durbin-Watson: 1.994
## Prob(Omnibus): 0.012 Jarque-Bera (JB): 11.283
## Skew: -0.012 Prob(JB): 0.00355
## Kurtosis: 3.328 Cond. No. 4.12e+03
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## [2] The condition number is large, 4.12e+03. This might indicate that there are
## strong multicollinearity or other numerical problems.
## AIC: 9470.09711553212
## BIC: 9505.048587280711
## Final Variables: ['intercept', 'LBXTR', 'BPXDI', 'AGE', 'HEIGHT', 'WEIGHT']
From the graphes above, we can see that LBXTR and AGE plots, there are a little inverse ‘U’ shape, so in the final model, we add LBXTR square and AGE square terms.
Data['AGE2']=Data[['AGE']]*Data[['AGE']]
Data['LBXTR2']=Data[['LBXTR']]*Data[['LBXTR']]
y5=Data[['LDL']]
x5=Data.drop(['LDL'],axis=1)
final_vars,_=forwardSelection(x5,y5,model_type='linear',elimination_criteria='aic')
## Character Variables (Dummies Generated, First Dummies Dropped): []
## Entered : LBXTR AIC : 9574.218891776083
## Entered : LBXTR2 AIC : 9448.966446031709
## Entered : BPXDI AIC : 9384.22136728393
## Entered : AGE AIC : 9373.946948986257
## Entered : AGE2 AIC : 9246.164944091768
## Break : Significance Level
## OLS Regression Results
## ==============================================================================
## Dep. Variable: LDL R-squared: 0.205
## Model: OLS Adj. R-squared: 0.203
## Method: Least Squares F-statistic: 128.6
## Date: Wed, 11 Dec 2019 Prob (F-statistic): 1.75e-121
## Time: 23:24:51 Log-Likelihood: -4617.1
## No. Observations: 2503 AIC: 9246.
## Df Residuals: 2497 BIC: 9281.
## Df Model: 5
## Covariance Type: nonrobust
## ==============================================================================
## coef std err t P>|t| [0.025 0.975]
## ------------------------------------------------------------------------------
## intercept 6.2939 0.195 32.322 0.000 5.912 6.676
## LBXTR 0.0213 0.002 13.348 0.000 0.018 0.024
## LBXTR2 -4.949e-05 4.9e-06 -10.094 0.000 -5.91e-05 -3.99e-05
## BPXDI 0.0078 0.003 2.971 0.003 0.003 0.013
## AGE 0.1004 0.008 11.994 0.000 0.084 0.117
## AGE2 -0.0010 9.04e-05 -11.528 0.000 -0.001 -0.001
## ==============================================================================
## Omnibus: 8.099 Durbin-Watson: 1.962
## Prob(Omnibus): 0.017 Jarque-Bera (JB): 9.982
## Skew: -0.024 Prob(JB): 0.00680
## Kurtosis: 3.306 Cond. No. 1.60e+05
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## [2] The condition number is large, 1.6e+05. This might indicate that there are
## strong multicollinearity or other numerical problems.
## AIC: 9246.164944091768
## BIC: 9281.116415840359
## Final Variables: ['intercept', 'LBXTR', 'LBXTR2', 'BPXDI', 'AGE', 'AGE2']
final_vars,_=backwardSelection(x5,y5,model_type='linear',elimination_criteria='aic')
## Character Variables (Dummies Generated, First Dummies Dropped): []
## Eliminated : FAT
## Eliminated : Other Hispanic
## Eliminated : Non-Hispanic Asian
## Eliminated : Female
## Eliminated : CHOL
## Eliminated : Non-Hispanic White
## Regained : Non-Hispanic White
## OLS Regression Results
## ==============================================================================
## Dep. Variable: LDL R-squared: 0.210
## Model: OLS Adj. R-squared: 0.207
## Method: Least Squares F-statistic: 55.30
## Date: Wed, 11 Dec 2019 Prob (F-statistic): 1.87e-118
## Time: 23:24:51 Log-Likelihood: -4608.2
## No. Observations: 2503 AIC: 9242.
## Df Residuals: 2490 BIC: 9318.
## Df Model: 12
## Covariance Type: nonrobust
## ======================================================================================
## coef std err t P>|t| [0.025 0.975]
## --------------------------------------------------------------------------------------
## intercept 11.3355 2.013 5.631 0.000 7.388 15.283
## LBXTR 0.0216 0.002 13.060 0.000 0.018 0.025
## BPXSY 0.0040 0.002 1.881 0.060 -0.000 0.008
## BPXDI 0.0065 0.003 2.279 0.023 0.001 0.012
## AGE 0.1063 0.009 12.338 0.000 0.089 0.123
## HEIGHT -0.0333 0.012 -2.731 0.006 -0.057 -0.009
## WEIGHT 0.0262 0.012 2.143 0.032 0.002 0.050
## BMI -0.0757 0.034 -2.228 0.026 -0.142 -0.009
## Non-Hispanic White 0.1253 0.075 1.667 0.096 -0.022 0.273
## Non-Hispanic Black 0.1642 0.088 1.876 0.061 -0.007 0.336
## Other Race 0.2850 0.167 1.702 0.089 -0.043 0.613
## AGE2 -0.0011 9.45e-05 -11.900 0.000 -0.001 -0.001
## LBXTR2 -5.047e-05 5e-06 -10.099 0.000 -6.03e-05 -4.07e-05
## ==============================================================================
## Omnibus: 7.701 Durbin-Watson: 1.963
## Prob(Omnibus): 0.021 Jarque-Bera (JB): 9.303
## Skew: -0.034 Prob(JB): 0.00955
## Kurtosis: 3.291 Cond. No. 1.66e+06
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## [2] The condition number is large, 1.66e+06. This might indicate that there are
## strong multicollinearity or other numerical problems.
## AIC: 9242.336445749102
## BIC: 9318.064634537715
## Final Variables: ['intercept', 'LBXTR', 'BPXSY', 'BPXDI', 'AGE', 'HEIGHT', 'WEIGHT', 'BMI', 'Non-Hispanic White', 'Non-Hispanic Black', 'Other Race', 'AGE2', 'LBXTR2']
From AIC comparison, we select variables ‘LBXTR’, ‘BPXSY’, ‘BPXDI’, ‘AGE’, ‘HEIGHT’, ‘WEIGHT’, ‘BMI’, ‘RACE’, ‘AGE2’, ‘LBXTR2’
y=Data[['LDL']]
x=Data[['LBXTR', 'BPXSY', 'BPXDI', 'AGE', 'HEIGHT', 'WEIGHT', 'BMI','Other Hispanic','Non-Hispanic White','Non-Hispanic Black','Non-Hispanic Asian','Other Race', 'AGE2', 'LBXTR2']]
#here we add all RACE types except RACE='Mexican American', which is viewed as base
x=sm.add_constant(x)
## /Users/lihuayu/anaconda3/lib/python3.7/site-packages/numpy/core/fromnumeric.py:2495: FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.
## return ptp(axis=axis, out=out, **kwargs)
lm_final=sm.OLS(y.astype(float),x.astype(float)).fit()
lm_final.summary()
## <class 'statsmodels.iolib.summary.Summary'>
## """
## OLS Regression Results
## ==============================================================================
## Dep. Variable: LDL R-squared: 0.211
## Model: OLS Adj. R-squared: 0.206
## Method: Least Squares F-statistic: 47.49
## Date: Wed, 11 Dec 2019 Prob (F-statistic): 5.38e-117
## Time: 23:24:51 Log-Likelihood: -4607.5
## No. Observations: 2503 AIC: 9245.
## Df Residuals: 2488 BIC: 9332.
## Df Model: 14
## Covariance Type: nonrobust
## ======================================================================================
## coef std err t P>|t| [0.025 0.975]
## --------------------------------------------------------------------------------------
## const 11.1998 2.017 5.552 0.000 7.244 15.155
## LBXTR 0.0216 0.002 13.012 0.000 0.018 0.025
## BPXSY 0.0041 0.002 1.922 0.055 -8.27e-05 0.008
## BPXDI 0.0063 0.003 2.212 0.027 0.001 0.012
## AGE 0.1057 0.009 12.245 0.000 0.089 0.123
## HEIGHT -0.0329 0.012 -2.700 0.007 -0.057 -0.009
## WEIGHT 0.0256 0.012 2.100 0.036 0.002 0.050
## BMI -0.0732 0.034 -2.150 0.032 -0.140 -0.006
## Other Hispanic 0.0896 0.111 0.810 0.418 -0.127 0.307
## Non-Hispanic White 0.1879 0.094 1.997 0.046 0.003 0.372
## Non-Hispanic Black 0.2247 0.103 2.175 0.030 0.022 0.427
## Non-Hispanic Asian 0.1355 0.125 1.081 0.280 -0.110 0.381
## Other Race 0.3472 0.177 1.966 0.049 0.001 0.694
## AGE2 -0.0011 9.46e-05 -11.824 0.000 -0.001 -0.001
## LBXTR2 -5.029e-05 5e-06 -10.053 0.000 -6.01e-05 -4.05e-05
## ==============================================================================
## Omnibus: 7.784 Durbin-Watson: 1.965
## Prob(Omnibus): 0.020 Jarque-Bera (JB): 9.359
## Skew: -0.038 Prob(JB): 0.00928
## Kurtosis: 3.290 Cond. No. 1.66e+06
## ==============================================================================
##
## Warnings:
## [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
## [2] The condition number is large, 1.66e+06. This might indicate that there are
## strong multicollinearity or other numerical problems.
## """
## Additional Analysis: Some extra graphs
## Libraries: -------------------------------------------------------------------------
library(ggplot2)
library(gridExtra)
## 80: --------------------------------------------------------------------------------
## Before graphing, we will have the linear mixed model upon the dataset:
LM=lmer(ldl~age+intake_fat+intake_chol+systolic+diastolic+
weight+height+bmi+triglycerides+(1|gender)+(1|race),data=DT)
summary(LM)
## Linear mixed model fit by REML ['lmerMod']
## Formula: ldl ~ age + intake_fat + intake_chol + systolic + diastolic +
## weight + height + bmi + triglycerides + (1 | gender) + (1 | race)
## Data: DT
##
## REML criterion at convergence: 24691.1
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.3525 -0.6773 -0.0739 0.5917 4.4911
##
## Random effects:
## Groups Name Variance Std.Dev.
## race (Intercept) 0.695 0.8337
## gender (Intercept) 0.000 0.0000
## Residual 1111.545 33.3398
## Number of obs: 2503, groups: race, 6; gender, 2
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 111.161885 43.863459 2.534
## age 0.175052 0.038036 4.602
## intake_fat 0.001994 0.022497 0.089
## intake_chol -0.001512 0.004350 -0.348
## systolic -0.009931 0.045057 -0.220
## diastolic 0.411477 0.056723 7.254
## weight 0.309732 0.265604 1.166
## height -0.351071 0.264700 -1.326
## bmi -0.630334 0.737863 -0.854
## triglycerides 0.141182 0.011105 12.714
##
## Correlation of Fixed Effects:
## (Intr) age intk_f intk_c systlc distlc weight height bmi
## age 0.073
## intake_fat 0.047 0.053
## intake_chol -0.011 -0.066 -0.596
## systolic -0.074 -0.458 0.027 0.015
## diastolic 0.011 0.102 -0.063 0.033 -0.302
## weight 0.961 0.071 0.017 -0.012 -0.009 0.022
## height -0.992 -0.054 -0.071 0.009 -0.003 -0.051 -0.963
## bmi -0.957 -0.081 -0.016 0.004 -0.003 -0.040 -0.991 0.955
## triglycerds -0.073 -0.134 0.053 -0.039 -0.047 -0.081 -0.093 0.077 0.067
## convergence code: 0
## boundary (singular) fit: see ?isSingular
According to the fitting result, the age, intake_fat, diastolic, weight and triglycerides are positively correlated, while the others are negatively correlated; the t-values of triglycerides, diastolic and age are really large, shows that they are important variables to this model. However, according to the random effects, the variance for race is 0.695, for gender is 0, and for residual is 1111.545, implies that the linear mixed model may not work really well.
## Graph 1: Some Diagnosis upon these models
### F1: Checking error assumptions--residual plots
R1=data.table(fitted_values=L2$fitted.values,residuals=L2$residuals)
R2=data.table(fitted_values=L3$fitted.values,residuals=L3$residuals)
R3=data.table(fitted_values=L4$fitted.values,residuals=L4$residuals)
R4=data.table(fitted_values=L5$fitted.values,residuals=L5$residuals)
rs1=ggplot(R1,aes(x=fitted_values,y=residuals))+geom_point(size=1,colour='blue')+
labs(title='Model 1')
rs2=ggplot(R2,aes(x=fitted_values,y=residuals))+geom_point(size=1,colour='blue')+
labs(title='Model 2')
rs3=ggplot(R3,aes(x=fitted_values,y=residuals))+geom_point(size=1,colour='blue')+
labs(title='Model 3')
rs4=ggplot(R4,aes(x=fitted_values,y=residuals))+geom_point(size=1,colour='blue')+
labs(title='Model 4')
grid.arrange(rs1,rs2,rs3,rs4,nrow=2)
## Graph 2: QQ-plots of the models
par(mfrow=c(2,2))
qqnorm(R1$residuals, ylab="Residuals",main='Q-Q Plot of Model 1')
qqline(R1$residuals)
qqnorm(R2$residuals, ylab="Residuals",main='Q-Q Plot of Model 2')
qqline(R2$residuals)
qqnorm(R3$residuals, ylab="Residuals",main='Q-Q Plot of Model 3')
qqline(R3$residuals)
qqnorm(R4$residuals, ylab="Residuals",main='Q-Q Plot of Model 4')
qqline(R4$residuals)
## Graph 3: Partial Residual Plots upon Model 3 and 4
crPlots(L4,layout=c(3,3))
crPlots(L5)
## Graph 4: Relationships between ldl and gender/race. We have the mean level of
## ldl between different gender and race (For CI, we will use the JackKnife
## standard error.)
Mean_JK = function(x){
lx=length(x)
MX=matrix(rep(x,rep(lx-1,lx)),ncol=lx,byrow=TRUE)
theta=colMeans(MX)
mean_theta=mean(theta)
std_theta={(lx-1)/lx*sum((theta-mean_theta)^2)}^(1/2)
std_theta
}
Gend=DT[,.(gender,ldl)]
Race=DT[,.(race,ldl)]
MG=Gend[,.(mean_ldl=mean(ldl),l_ldl=mean(ldl)+qnorm(0.025)*Mean_JK(ldl),
r_ldl=mean(ldl)+qnorm(0.975)*Mean_JK(ldl)),by=gender]
MR=Race[,.(mean_ldl=mean(ldl),l_ldl=mean(ldl)+qnorm(0.025)*Mean_JK(ldl),
r_ldl=mean(ldl)+qnorm(0.975)*Mean_JK(ldl)),by=race]
gend=ggplot(Gend,aes(x=gender,y=ldl))+geom_point(size=1,colour='blue')+
labs(title='ldl~gender')
race=ggplot(Race,aes(x=race,y=ldl))+geom_point(size=1,colour='blue')+
labs(title='ldl~race')
mean_gend=ggplot(MG,aes(x=gender,y=mean_ldl))+geom_point(shape=16,col='red')+
geom_segment(data=MG,mapping=aes(x=gender,xend=gender,y=l_ldl,yend=r_ldl),
col='blue')+
labs(title = 'Mean level of ldl between genders')
mean_race=ggplot(MR,aes(x=race,y=mean_ldl))+geom_point(shape=16,col='red')+
geom_segment(data=MR,mapping=aes(x=race,xend=race,y=l_ldl,yend=r_ldl),
col='blue')+
labs(title = 'Mean level of ldl between races')
grid.arrange(gend,race,mean_gend,mean_race,nrow=2)
Here we plot the residual plots and the QQ-plots, and the result shows that the regression models satisfy the assumptions of the OLS model. By the way, we plot the relationships between ldl and gender and race, and the plots shows that the ldl level of male is slightly higher than female from the whole, and the levels for female are more centerized. For race, the ldl level of Other Race is the highest from the whole, and Non-Hispanic Black is the lowest.
I am interested in exploring the specific relationship between the level of triglyceride and the response varaiable \(\sqrt{ldl}\). In partcular, does the relationship varies with the specific quantile of triglyceride? I used quantile regression to find the answer.
qreg ldl2 triglyceride, quantile(0.05)
qreg ldl2 triglyceride, quantile(0.25)
qreg ldl2 triglyceride, quantile(0.5)
qreg ldl2 triglyceride, quantile(0.75)
qreg ldl2 triglyceride, quantile(0.90)
Quantile Regression Results
Quantile Regression Plot
The quantile plot above shows that the level of triglyceride has positive correlation with the \(\sqrt{ldl}\) and the relationship between the two doesn’t vary much by quantile. Also, note that the quantile plot is generated with R and a detailed script is located under the folder /Xiru Lyu/quantile.R.
# additional analysis
#quantile regression
mod=smf.quantreg('LDL~AGE',Data)
quantiles = np.arange(.05, .96, .1)
def fit_model(q): # learned seaborn from and this function origin from https://www.cnblogs.com/caiyishuai/p/11184166.html
res = mod.fit(q=q)
return [q, res.params['Intercept'], res.params['AGE']] + \
res.conf_int().loc['AGE'].tolist()
models = [fit_model(x) for x in quantiles]
models = pd.DataFrame(models, columns=['q', 'a', 'b', 'lower_b', 'upper_b'])
ols = smf.ols('LDL ~ AGE', Data).fit()
ols_ci = ols.conf_int().loc['AGE'].tolist()
ols = dict(a = ols.params['Intercept'],
b = ols.params['AGE'],
lower_b = ols_ci[0], # Ci for b,lowerbound
upper_b = ols_ci[1]) #CI for b, upbound
print(models)
## q a b lower_b upper_b
## 0 0.05 7.365460 0.003383 -0.004561 0.011326
## 1 0.15 8.134052 0.008011 0.003168 0.012855
## 2 0.25 8.476658 0.013114 0.008268 0.017960
## 3 0.35 8.848354 0.016674 0.012116 0.021231
## 4 0.45 9.222907 0.017595 0.013548 0.021642
## 5 0.55 9.687246 0.016327 0.012471 0.020183
## 6 0.65 9.993639 0.019470 0.015664 0.023275
## 7 0.75 10.422161 0.020762 0.016958 0.024566
## 8 0.85 11.041352 0.020834 0.016755 0.024912
## 9 0.95 12.041026 0.022692 0.017059 0.028326
print(ols)
## {'a': 9.590565977646861, 'b': 0.014434702126817935, 'lower_b': 0.011225620121011554, 'upper_b': 0.017643784132624317}
x = np.arange(Data.AGE.min(), Data.AGE.max()+1, 1)
get_y = lambda a, b: a + b * x
fig, ax = plt.subplots(figsize=(8, 6))
for i in range(models.shape[0]):
y = get_y(models.a[i], models.b[i])
ax.plot(x, y, linestyle='dotted', color='g')
## [<matplotlib.lines.Line2D object at 0x11f282588>]
## [<matplotlib.lines.Line2D object at 0x12a851710>]
## [<matplotlib.lines.Line2D object at 0x12a92f978>]
## [<matplotlib.lines.Line2D object at 0x12a92f908>]
## [<matplotlib.lines.Line2D object at 0x12a956400>]
## [<matplotlib.lines.Line2D object at 0x12a956940>]
## [<matplotlib.lines.Line2D object at 0x12a956128>]
## [<matplotlib.lines.Line2D object at 0x12a9560b8>]
## [<matplotlib.lines.Line2D object at 0x12a956860>]
## [<matplotlib.lines.Line2D object at 0x12a94c4e0>]
y = get_y(ols['a'], ols['b'])
ax.plot(x, y, color='red', label='OLS')
## [<matplotlib.lines.Line2D object at 0x12a94c5c0>]
ax.scatter(Data.AGE, Data.LDL, alpha=.2)
## <matplotlib.collections.PathCollection object at 0x12a94c518>
ax.set_xlim((10, 85))
## (10, 85)
ax.set_ylim((3.0, 17.5))
## (3.0, 17.5)
legend = ax.legend()
ax.set_xlabel('AGE', fontsize=16)
## Text(0.5, 0, 'AGE')
ax.set_ylabel('LDL(sqrt_LDL)', fontsize=16);
## Text(0, 0.5, 'LDL(sqrt_LDL)')
Quantile reg: sqrt(ldl)~age
All of the three tools above use the stepwise model selection technique to choose the best fitted model, but there are some differences existing among these tools.
For the tool using data.table in R, the model selection function is just as step(,direction=''), and the direction choose is both; note that the selection criterion is by AIC. For Stata, the model selection technique is performed using stepwise, and the selection criterion is based on p-values. As Stata cannot perform the stepwise selection in both directions, separate forward and backward selections are performed, and p-values are tuned so that model selection results match those by R. For Python, the code for backward and forward selections are downloaded from the given Github site, and the selection criteria is by AIC. As the model selection technique differs for Python in selecting categorical variables of multiple levels, the final regression results by Python are slightly different from those by R and Stata.
Note for Python, we use pandas to merge and clean all the dataset.Because there is no package about step regression to help select variables, so we refer to some self-written forward step regression function based on adjusted R-square to select variables. For GLM regression, we import module sklearn.linear_model.LinearRegression. As for the result, to accord with my partners, when choose variables ‘LBXTR’(triglycerides level), ‘BPXSY’(systolic blood pressure), ‘BPXDI’(diastolic blood pressure), ‘FAT’(average fat intake), ‘CHOL’(average Cholesterol intake), ‘GENDER’, ‘AGE’,‘RACE’, ‘HEIGHT’, ‘WEIGHT’, ‘BMI’,‘LBXTR2’(LBXTRLBXTR), ‘AGE2’(AGEAGE), I got the same result, the forward select function showed me that the significant variables should include LBXTR LBXTR2 BPXDI AGE AGE2 RACE BPXSY HEIGHT CHOL with adjusted R-square being 0.20377. After regression within LinearRegression module, the regression R-square was 0.207806, which means these data can express around 20% of cholesterol level change. Correspondingly, the coefficients of these variables are 4.37936363e-01, -9.94588617e-04, 1.12071396e-01,2.09660709e+00, -2.20899498e-02, 9.69819194e-01, 8.32880449e-02, -1.07237225e-01, -3.73454180e-03.
However it seems that in python, there is no good way to set a variable factor, such as race. So to fix this problem, I used mixed model in module statsmodel, and the result showed that coefficients be:intercept 38.613, LBXTR 0.436, LBXTR2 -0.001,BPXDI 0.119, AGE 2.101, AGE2 -0.022, BPXSY 0.080, HEIGHT -0.108, CHOL -0.004, and Group Var 1.897.
And from the result, we can notice that first gender has no effect on cholesterol level(mg/dl), second height to some extent affect the level rather than weight, though we may have intuition that fatter people may have more cholesterol amount, it does not change the cholesterol density(mg/dl) and maybe that is why variable fat intake here barely have impactions. Besides though we might consider that intake will increase the relevant material level, here we see that cholesterol average intake in two days decrease the density of blood cholesterol, it might be because that two day records are not representative for a long term intake and adjust system may react to food intake in a short time just like blood glucose level in normal life, and the p-value for CHOL is the biggest 0.258 which means its true value could be zero. Third, we can notice that ‘BPXDI’(diastolic blood pressure) impact more than ‘BPXSY’(systolic blood pressure). Forth, with age increasing, the cholesterol density(mg/dl) will increase by 2mg/dl per year, but the increase rate will slow down for coefficient of AGE2 is negative.